DOCSLIB.ORG

Sparse Days September 6Th – 8Th, 2017 CERFACS, Toulouse, France

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Sparse Days September 6Th – 8Th, 2017 CERFACS, Toulouse, France Sparse Days September 6th – 8th, 2017 CERFACS, Toulouse, France Sparse Days – September 6th – 8th, 2017 – CERFACS Preliminary program Wednesday, September 6th 2017 13:00 – 14:00 registration 14:00 – 14:15 welcome message Session 1 (14:15 – 15:30) 14:15 – 15:40 Block Low-Rank multifrontal solvers: complexity, performance, and scalability Theo Mary (IRIT) 15:40 – 16:05 Sparse Supernodal Solver exploiting Low-Rankness Property Grégoire Pichon (Inria Bordeaux – Sud-Ouest) 15:05 – 15:30 Recursive inverse factorization Anton Artemov (Uppsala University) Coffee break (15:30 – 16:00) Session 2 (16:00 – 17:15) 16:00 – 16:25 Algorithm based on spectral analysis to detect numerical blocks in matrices Luce le Gorrec (IRIT – APO) 16:25 – 16:50 Computing Sparse Tensor Decompositions using Dimension Trees Oguz Kaya (ENS Lyon) 16:50 – 17:15 Numerical analysis of dynamic centrality Philip Knight (University of Strathclyde) Thursday, September 7th 2017 Session 3 (09:45 – 11:00) 09:45 – 10:10 Parallel Algorithm Design via Approximation Alex Pothen (Purdue University) 10:10 – 10:35 Accelerating the Mondriaan sparse matrix partitioner Rob Bisseling (Utrecht University) 10:35 – 11:00 LS-GPart: a global, distributed ordering library François-Henry Rouet (Livermore Software Technology Corporation) Coffee break (11:00 – 11:25) Session 4 (11:25 – 13:05) 11:25 – 11:50 A Tale of Two Codes Robert Lucas (Livermore Software Technology Corportion) 11:50 – 12:15 Direct solution of sparse systems of linear equations with sparse multiple right hand sides Gilles Moreau (ENS Lyon) 12:15 – 12:40 Averaged information splitting and multi-frontal solvers for heterogeneous high-throughput data analysis Shengxin Zhu (Xi'an Jiaotong – Liverpool University) 12:40 – 13:05 Permuting Spiked Matrices to Triangular Form in the Linear Programming LU Update Lukas Schork (University of Edinburgh, UK) Lunch break (13:05 – 15:00) Session 5 (15:00 – 16:15) 15:00 – 15:25 Factorization Based Sparse Solvers and Preconditioners for Exascale Sherry Li (Lawrence Berkeley National Laboratory) 15:25 – 15:50 Scalability Analysis of Sparse Matrix Computations on Many-core Processors Weifeng Liu (Norwegian University of Science and Technology) 15:50 – 16:15 Refactoring Sparse Triangular Solver on Sunway TaihuLight Many-core Supercomputer Wei Xue (Tsinghua University) Coffee break (16:15 – 16:45) 16:45 – 17:30 Review of Sparse Days by Iain Duff. Reception dinner (19:45) Friday, September 8th 2017 Session 6 (09:15 – 10:55) 09:15 – 09:40 On solving mixed sparse-dense linear least-squares problems Part I: A Schur complement approach 09:40 – 10:05 Part II: A preconditioned conjugate gradient method Jennifer Scott (STFC Rutherford Appleton Laboratory); and Miroslav Tuma (Charles University and Czech Academy of Sciences) 10:05 – 10:30 A Parallel Hierarchical Low-Rank Solver for General Sparse Matrices Erik Boman (Sandia National Laboratories) 10:30 – 10:55 Effective and robust SIF preconditioners for general SPD sparse matrices Jianlin Xia (Purdue University) Coffee break (10:55 – 11:20) Session 7 (11:20 – 13:00) 11:20 – 11:45 High Performance Block Incomplete LU Factorization Matthias Bollhoefer (TU Braunschweig) 11:15 – 12:10 An optimal Q-OR method for solving nonsymmetric linear systems Gérard Meurant 12:10 – 12:35 Preconditioning for non-symmetric Toeplitz matrices with application to time-dependent PDEs Andrew Wathen (Oxford University) 12:35 – 13:00 Efficient Parallel iterative solvers for high order DG methods Matthias Maischak (Brunel University) Lunch (13:00) Recursive inverse factorization Anton Artemov (Uppsala University, Sweden) In this talk we will present a parallel algorithm for the inverse factorization of symmetric positive definite matrices. This algorithm utilizes a hierarchical representation of the matrix and computes an inverse factor by combining an iterative refinement procedure with a recursive binary principal submatrix decomposition. Although our algorithm should be generally applicable, this work was mainly motivated by the need for efficient and scalable inverse factorization of the basis set overlap matrix in large scale electronic structure calculations. We show that for such matrices the computational cost increases only linearly with system size. We will also demonstrate the weak scaling behavior of the algorithm and give theoretical justification of the scaling behavior based on critical path length estimation. Accelerating the Mondriaan sparse matrix partitioner Rob Bisseling (Mathematical Institute, Utrecht University, Netherlands) The Mondriaan package for sparse matrix partitioning has been developed since 2002 and currently it has reached version 4.1. Its latest addition is the medium-grain method for 2D partitioning, which significantly reduces the communication volume of a parallel sparse matrix-vector multiplication. The present talk is about recent developments to be included in version 4.2 of Mondriaan, which focus on improving the speed of computing the partitioning. One improvement is the use of free nonzeros in the matrix, which can be moved after a split to improve the load balance without adversely affecting the communication volume. This leaves more room for reducing communication in subsequent splits. Another improvement for certain sparse matrices is obtained by trying to find connected components after every split that satisfy the imposed imbalance constraint. If this succeeds, a zero-volume split is found very quickly. The total savings in computation time of these and other improvement for a set of 900 sparse matrices is about 25 per cent. This is joint work with Marco van Oort High Performance Block Incomplete LU Factorization Matthias Bollhoefer (TU Braunschweig, Germany) Many application problems that lead to solving linear systems make use of preconditioned Krylov subspace solvers to compute their solution. Among the most popular preconditioning approaches are incomplete factorization methods either as single--level approaches or within a multilevel framework. We will present a block incomplete triangular factorization that is based on skillfully blocking the system initially and throughout the factorization. This approach allows for the use of cache-optimized dense matrix kernels such level-3 BLAS or LAPACK. We will demonstrate how this block approach may significantly outperform the scalar method on modern architectures, paving the way for its prospective use inside various multilevel incomplete factorization approaches or other applications where the core part relies on an incomplete factorization. A Parallel Hierarchical Low-Rank Solver for General Sparse Matrices Erik Boman (Sandia National Laboratories, USA) Hierarchical methods that exploit low-rank structure have attracted much attention lately. For sparse matrices, many methods rely on low-rank structure within dense frontal matrices. We present a different approach by Eric Darve et al. that works on the entire sparse matrix, and is in several ways simpler. The method is most useful as a preconditioner, though it could also be used as a direct solver. We show that the method can be viewed as an H2 solver and also as an extension to the block ILU(0) preconditioner. We briefly discuss a parallel code developed in collaboration with Stanford, and show some results. Joint work with Chao Chen, Eric Darve, Siva Rajamanickam, and Ray Tuminaro. Computing Sparse Tensor Decompositions using Dimension Trees Oguz Kaya (ENS Lyon, France) Tensor decompositions have increasingly been used in many applications domains including computer vision, signal processing, graph analytics, healthcare data analysis, recommender systems, and many other related machine learning problems. In many of these applications, the tensor representation of data is big, sparse, and of low-rank, and an efficient computation of tensor decompositions is indispensible to be able to analyze datasets of massive scale. Two major computational kernels involved at the core of the iterative tensor decomposition algorithms are tensor- times-vector (TTV) and -matrix (TTM) multiplications. In this talk, we propose a novel algorithmic framework along with an associated sparse tensor data structure that enables an efficient computation of these two operations for high dimensional sparse tensors. With this framework, we enable storing and reusing partial TTV and TTM results, thereby reducing the cost per iteration to O(NlogN) TTM/TTV calls from O(N^2) in the traditional scheme. Second, we propose an efficient tensor storage scheme representing an N-dimensional tensor using O(NlogN) index arrays, and enabling the partial computations in this framework. Third, we propose shared memory parallelization of TTV and TTMs using this sparse data representation. With all these optimizations, we up to 6x speedup over the state of the art implementations using 32 dimensional sparse tensors. Numerical analysis of dynamic centrality Philip Knight (University of Strathclyde, UK) Centrality measures have proven to be a vital tool for analysing static networks. In recent years, many of these measures have been adapted for use on dynamic networks. In particular, spectral measures such as Katz centrality have been extended to take into account time's arrow. We show that great care must be taken in applying such measures due to the inherent ill-conditioning in the associated matrices (an ill-conditioning which can get so extreme that the matrices involved can have a numerical rank of 1). At the same time, the ill-conditioning can reduce computational
Load more

Recommended publications
  • 3.1 Ducom-COM3
    Addis Ababa University Addis Ababa Institute of Technology School of Electrical and Computer Engineering Performance Improvement of Multi-scale Modeling of Concrete Performance Solver Using Hybrid CPU-GPU System A Thesis Submitted to the School of Electrical and Computer Engineering In Partial Fulfillment of the Requirements for the Degree of Master of Science In Computer Engineering By: Lina Felleke Advisor: Fitsum Assamnew Co-Advisor: Dr. Esayas G/Yohannes April 2015 Addis Ababa University Addis Ababa Institute of Technology School of Electrical and Computer Engineering Performance Improvement of Multi-scale Modeling of Concrete Performance Solver Using Hybrid CPU-GPU System By: Lina Felleke Advisor: Fitsum Assamnew Co-Advisor: Dr. Esayas G/Yohannes April 2015 Addis Ababa University Addis Ababa Institute of Technology School of Electrical and Computer Engineering Performance Improvement of Multi-scale Modeling of Concrete Performance Solver Using Hybrid CPU-GPU System By Lina Felleke Approval by Board of Examiners Dr.Yalemzewd Negash Dean, School of Electrical Signature And Computer Engineering Fitsum Assamnew Advisor Signature Dr. Esayas_G/Yohannes Co-Advisor Signature Dr. Sirinavas Nune ________________ External Examiner Signature Menore Tekeba _________________ Internal Examiner Signature i Declaration I, the undersigned, declare that this thesis work, to the best of my knowledge and belief, is my original work, has not been presented for a degree in this or any other universities, and all sources of materials used for the thesis work have been fully acknowledged. Name: LinaFelleke Signature: _________________ Place: Addis Ababa Date of submission: This thesis has been submitted for examination with my approval as a university advisor. Fitsum Assamnew Signature: ___________________ Advisor Name Dr.
    [Show full text]
  • Sparse Days September 6Th – 8Th, 2017 CERFACS, Toulouse, France
    Sparse Days September 6th – 8th, 2017 CERFACS, Toulouse, France Sparse Days programme Wednesday, September 6th 2017 13:00 – 14:00 Registration 14:00 – 14:15 Welcome message Session 1 (14:15 – 15:30) – Chair: Iain Duff 14:15 – 15:40 Block Low-Rank multifrontal solvers: complexity, performance, and scalability Theo Mary (IRIT) 15:40 – 16:05 Sparse Supernodal Solver exploiting Low-Rankness Property Grégoire Pichon (Inria Bordeaux – Sud-Ouest) 15:05 – 15:30 Recursive inverse factorization Anton Artemov (Uppsala University) Coffee break (15:30 – 16:00) Session 2 (16:00 – 17:40) – Chair: Daniel Ruiz 16:00 – 16:25 Algorithm based on spectral analysis to detect numerical blocks in matrices Luce le Gorrec (IRIT – APO) 16:25 – 16:50 Computing Sparse Tensor Decompositions using Dimension Trees Oguz Kaya (ENS Lyon) 16:50 – 17:15 Numerical analysis of dynamic centrality Philip Knight (University of Strathclyde) 17:15 – 17:40 SuiteSparse:GraphBLAS: graph algorithms via sparse matrix operations on semirings Tim Davis (Texas A&M University) Thursday, September 7th 2017 Session 3 (09:45 – 11:00) – Chair: Tim Davis 09:45 – 10:10 Parallel Algorithm Design via Approximation Alex Pothen (Purdue University) 10:10 – 10:35 Accelerating the Mondriaan sparse matrix partitioner Rob Bisseling (Utrecht University) 10:35 – 11:00 LS-GPart: a global, distributed ordering library François-Henry Rouet (Livermore Software Technology Corporation) Coffee break (11:00 – 11:25) Session 4 (11:25 – 13:05) – Chair: Cleve Ashcraft 11:25 – 11:50 A Tale of Two Codes Robert Lucas
    [Show full text]
  • A Block Sparse Shared-Memory Multifrontal Finite Element Solver For
    Computer Assisted Mechanics and Engineering Sciences, 16 : 117–131, 2009. Copyright c 2009 by Institute of Fundamental Technological Research, Polish Academy of Sciences A block sparse shared-memory multifrontal finite element solver for problems of structural mechanics S. Y. Fialko Institute of Computer Modeling, Cracow University of Technology ul. Warszawska 24, 31-155 Cracow, Poland Software Company SCAD Soft, Ukraine (Received in the final form July 17, 2009) The presented method is used in finite-element analysis software developed for multicore and multipro- cessor shared-memory computers, or it can be used on single-processor personal computers under the operating systems Windows 2000, Windows XP, or Windows Vista, widely popular in small or medium- sized design offices. The method has the following peculiar features: it works with any ordering; it uses an object-oriented approach on which a dynamic, highly memory-efficient algorithm is based; it performs a block factoring in the frontal matrix that entails a high-performance arithmetic on each processor and ensures a good scalability in shared-memory systems. Many years of experience with this solver in the SCAD software system have shown the method’s high efficiency and reliability with various large-scale problems of structural mechanics (hundreds of thousands to millions of equations). Keywords: finite element method, large-scale problems, multifrontal method, sparse matrices, ordering, multithreading. 1. I NTRODUCTION The dimensionality of present-day design models for multistorey buildings and other engineering structures may reach 1,000,000 to 2,000,000 equations and still grows up. Therefore, one of most important issues with modern CAD software is how fast and reliably their solver subsystems can solve large-scale problems.
    [Show full text]
  • A Survey of Direct Methods for Sparse Linear Systems
    A survey of direct methods for sparse linear systems Timothy A. Davis, Sivasankaran Rajamanickam, and Wissam M. Sid-Lakhdar Technical Report, Department of Computer Science and Engineering, Texas A&M Univ, April 2016, http://faculty.cse.tamu.edu/davis/publications.html To appear in Acta Numerica Wilkinson defined a sparse matrix as one with enough zeros that it pays to take advantage of them.1 This informal yet practical definition captures the essence of the goal of direct methods for solving sparse matrix problems. They exploit the sparsity of a matrix to solve problems economically: much faster and using far less memory than if all the entries of a matrix were stored and took part in explicit computations. These methods form the backbone of a wide range of problems in computational science. A glimpse of the breadth of applications relying on sparse solvers can be seen in the origins of matrices in published matrix benchmark collections (Duff and Reid 1979a) (Duff, Grimes and Lewis 1989a) (Davis and Hu 2011). The goal of this survey article is to impart a working knowledge of the underlying theory and practice of sparse direct methods for solving linear systems and least-squares problems, and to provide an overview of the algorithms, data structures, and software available to solve these problems, so that the reader can both understand the methods and know how best to use them. 1 Wilkinson actually defined it in the negation: \The matrix may be sparse, either with the non-zero elements concentrated ... or distributed in a less systematic manner. We shall refer to a matrix as dense if the percentage of zero elements or its distribution is such as to make it uneconomic to take advantage of their presence." (Wilkinson and Reinsch 1971), page 191, emphasis in the original.
    [Show full text]
  • Numerical Analysis Group Progress Report
    RAL 94-062 NUMERICAL ANALYSIS GROUP PROGRESS REPORT January 1991 ± December 1993 Edited by Iain S. Duff Central Computing Department Atlas Centre Rutherford Appleton Laboratory Oxon OX11 0QX June 1994. CONTENTS 1 Introduction (I.S. Duff) ¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼ 2 2 Sparse Matrix Research ¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼ 6 2.1 The solution of sparse structured symmetric indefinite linear sets of equations (I.S. Duff and J.K. Reid)¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼ 6 2.2 The direct solution of sparse unsymmetric linear sets of equations (I.S. Duff and J.K. Reid) ¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼ 7 2.3 Development of a new frontal solver (I. S. Duff and J. A. Scott) ¼¼¼¼¼¼¼ 10 2.4 A variable-band linear equation frontal solver (J. A. Scott) ¼¼¼¼¼¼¼¼¼ 12 2.5 The use of multiple fronts (I. S.Duff and J. A. Scott)¼¼¼¼¼¼¼¼¼¼¼¼ 13 2.6 MUPS: a MUltifrontal Parallel Solver for sparse unsymmetric sets of linear equations (P.R. Amestoy and I.S. Duff)¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼ 14 2.7 Sparse matrix factorization on the BBN TC2000 (P.R. Amestoy and I.S. Duff) ¼ 15 2.8 Unsymmetric multifrontal methods (T.A. Davis and I.S. Duff) ¼¼¼¼¼¼¼ 17 2.9 Unsymmetric multifrontal methods for finite-element problems (A.C. Damhaug, I.S. Duff, J.A. Scott, and J.K. Reid) ¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼ 18 2.10 Automatic scaling of sparse symmetric matrices (J.K. Reid) ¼¼¼¼¼¼¼¼ 19 2.11 Block triangular form and QR decomposition (I.S. Duff and C. Puglisi)¼¼¼¼ 20 2.12 Multifrontal QR factorization in a multiprocessor environment (P.R. Amestoy, I.S. Duff, and C. Puglisi) ¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼ 22 2.13 Impact of full QR factorization on sparse multifrontal QR algorithm (I.S.
    [Show full text]
  • Towards Green Multi-Frontal Solver for Adaptive Finite Element Method
    See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/277587946 Towards Green Multi-frontal Solver for Adaptive Finite Element Method Article in Procedia Computer Science · June 2015 DOI: 10.1016/j.procs.2015.05.240 CITATION READS 1 29 6 authors, including: Konrad Jopek Pawel Gepner AGH University of Science and Technology in … Intel 13 PUBLICATIONS 37 CITATIONS 42 PUBLICATIONS 288 CITATIONS SEE PROFILE SEE PROFILE Jacek Kitowski Maciej Paszynski AGH University of Science and Technology in … AGH University of Science and Technology in … 228 PUBLICATIONS 851 CITATIONS 178 PUBLICATIONS 1,282 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Solver performance analysis and optimization View project Linear computational cost and memory usage hybrid solvers for problems of electromagnetic waves propagation over the human head model View project All content following this page was uploaded by Maciej Paszynski on 02 June 2015. The user has requested enhancement of the downloaded file. Procedia Computer Science Volume 51, 2015, Pages 984–993 ICCS 2015 International Conference On Computational Science Towards green multi-frontal solver for adaptive finite element method H. AbbouEisha1,M.Moshkov1,K.Jopek2,P.Gepner3, J. Kitowski2,andM. Paszy´nski2 1 King Abdullah University of Science and Technology, Thuwal, Saudi Arabia 2 AGH University of Science and Technology, Krakow, Poland 3 Intel Corporation, Swindon, United Kingdom Abstract In this paper we present the optimization of the energy consumption for the multi-frontal solver algorithm executed over two dimensional grids with point singularities. The multi-frontal solver algorithm is controlled by so-called elimination tree, defining the order of elimination of rows from particular frontal matrices, as well as order of memory transfers for Schur complement matrices.
    [Show full text]
  • A Fast Direct Solver for Structured Linear Systems by Recursive Skeletonization ∗
    SIAM J. SCI. COMPUT. c 2012 Society for Industrial and Applied Mathematics Vol. 34, No. 5, pp. A2507–A2532 A FAST DIRECT SOLVER FOR STRUCTURED LINEAR SYSTEMS BY RECURSIVE SKELETONIZATION ∗ † ‡ KENNETH L. HO AND LESLIE GREENGARD Abstract. We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the original matrix into a larger but highly structured sparse one that allows fast factorization and application of the inverse. The algorithm extends the Martinsson–Rokhlin method developed for 2D boundary integral equations and proceeds in two phases: a precomputation phase, consisting of matrix compression and factorization, followed by a solution phase to apply the matrix inverse. For boundary integral equations which are not too oscillatory, e.g., based on the Green functions for the Laplace or low-frequency Helmholtz equa- O N N tions, both phases typically have complexity ( ) in two dimensions, where is the number of discretization points. In our current implementation, the corresponding costs in three dimensions are O(N 3/2)andO(N log N) for precomputation and solution, respectively. Extensive numerical exper- iments show a speedup of ∼100 for the solution phase over modern fast multipole methods; however, the cost of precomputation remains high. Thus, the solver is particularly suited to problems where large numbers of iterations would be required. Such is the case with ill-conditioned linear systems or when the same system is to be solved with multiple right-hand sides.
    [Show full text]
  • Parallel Sparse Direct Methods: a Short Tutorial
    RC 25076, November 10, 2010 Computer Science/Mathematics IBM Research Report Parallel Sparse Direct Methods: A short tutorial Anshul Gupta Business Analytics and Mathematical Sciences IBM T. J. Watson Research Center [email protected] LIMITED DISTRIBUTION NOTICE A shorter version of this report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g., payment of royalties). Research Division IBM Almaden Austin China Haifa India Tokyo Watson Zurich Parallel Sparse Direct Methods: A short tutorial Anshul Gupta Business Analytics and Mathematical Sciences IBM T. J. Watson Research Center Yorktown Heights, NY 10598 [email protected] November 10, 2010 1 Definition ¡¨¤§© © Direct methods for solving linear systems of the form ¡£¢¥¤§¦ are based on computing , where and are lower and upper triangular, respectively. Computing the triangular factors of the coefficient matrix ¡ is also known as LU decomposition. Following the factorization, the original system is trivially solved by ¢¤ ¡ solving the triangular systems © ¤¦ and . If is symmetric, then a factorization of the form ¡¤©£© © ¡¤©£© or is computed via Cholesky factorization, where is a lower triangular matrix (unit lower triangular in the case of ¡¤©£© factorization) and is a diagonal matrix. One set of common formulations of LU decomposition and Cholesky factorization for dense matrices are shown in Figures 1 and 2, respectively.
    [Show full text]
  • Solver Schemes for Linear Systems Oral Comprehensive Exam Position Paper
    Solver Schemes for Linear Systems Oral Comprehensive Exam Position Paper Kanika Sood Department of Computer and Information Science University of Oregon December 9, 2016 1 Introduction This report presents different approaches to solving sparse linear systems|direct and iterative solvers|and compares a number of methods in each category. I also consider the two techniques of using solvers for linear systems: single method solver schemes and multi-method solver schemes. I also present some of the popular systems that use these techniques. This involves surveying the relevant literature that contains in-depth information about techniques in the field. Linear systems are a form of representation for problems in a variety of domains, including but not limited to statistics, thermodynamics, electric circuits, quantum mechanics, nuclear engi- neering, fossil fuels, robotics and computational fluid dynamics. Solution methods in numerical optimization and PDE-based simulations frequently rely on the efficient solution of large, typically sparse linear systems. Given that linear systems are widespread [75, 74] across different areas of research, providing accurate and efficient solution methods plays a critical role for scientists in these fields. There are two classes of linear systems: sparse and dense. The systems in which most of the elements are zero are known as sparse systems. Those systems for which most of the elements are non-zero are referred to as dense linear systems. A common approach of storing sparse matrices requires only the non-zero elements to be stored, along with their location, instead of storing all the elements including the zero elements. Large sparse linear systems tend to represent real world problems better than dense systems and large sparse systems occur more frequently than small or dense systems.
    [Show full text]
  • Grammar Based Multi-Frontal Solver for Isogeometric Analysis in 1D
    Computer Science 14 (4) 2013 http://dx.doi.org/10.7494/csci.2013.14.4.589 • Krzysztof Kuźnik Maciej Paszyński Victor Calo GRAMMAR BASED MULTI-FRONTAL SOLVER FOR ISOGEOMETRIC ANALYSIS IN 1D Abstract In this paper, we present a multi-frontal direct solver for one-dimensional iso- geometric finite element method. The solver implementation is based on the graph grammar (GG) model. The GG model allows us to express the entire solver algorithm, including generation of frontal matrices, merging, and elimi- nations as a set of basic undividable tasks called graph grammar productions. Having the solver algorithm expressed as GG productions, we can find the par- tial order of execution and create a dependency graph, allowing for scheduling of tasks into shared memory parallel machine. We focus on the implementation of the solver with NVIDIA CUDA on the graphic processing unit (GPU). The solver has been tested for linear, quadratic, cubic, and higher-order B-splines, resulting in logarithmic scalability. Keywords graph grammar, direct solver, isogeometric finite element method, NVIDIA CUDA GPU 589 1 lutego 2014 str. 1/25 590 Krzysztof Kuźnik, Maciej Paszyński, Victor Calo 1. Introduction The finite element method with hierarchical shape functions [6, 7] delivers higher or- der polynomials, but the continuity of the global approximation is only C0 between particular mesh elements. This was the main motivation for developing the isogeome- tric finite element method [5], which utilizes the B-splines as basis functions and, thus, delivers Ck global continuity. The multi-frontal direct solver is a state-of-the art algorithm for solving sparse linear systems of equations generated by finite element discretizations [9, 13].
    [Show full text]
  • High Performance Block Incomplete LU Factorization
    High Performance Block Incomplete LU Factorization Matthias Bollh¨ofer ∗ Olaf Schenk y Fabio Verbosio z August 28, 2019 Abstract Many application problems that lead to solving linear systems make use of preconditioned Krylov subspace solvers to compute their solution. Among the most popular preconditioning approaches are incomplete factorization methods either as single-level approaches or within a multilevel framework. We will present a block incomplete factorization that is based on skillfully blocking the system initially and throughout the factorization. This approach allows for the use of cache-optimized dense matrix kernels such as level-3 BLAS or LAPACK. We will demonstrate how this block approach outperforms the scalar method often by orders of magnitude on modern architectures, paving the way for its prospective use inside various multilevel incomplete factorization approaches or other applications where the core part relies on an incomplete factorization. Keywords: sparse matrices, incomplete LU factorizations, block-structured methods, dense matrix kernels, block ILU. 1 Introduction Many application problems lead to solving linear systems of type Ax = b; where A is an n × n nonsingular real or complex system matrix and b is the associated right-hand side. In particular we are interested in the case where A is large-scale and sparse. The generic way of solving these systems nowadays consists of using state-of-the-art sparse direct solvers (cf., e.g., [3, 40, 10, 31]). Although high performance sparse direct solvers are very efficient in many cases, several structured problems, i.e., problems presenting specific, noticeable sparsity structures, cause the direct solver to produce a significant amount of fill-in during the factorization, leading to high memory requirements which can exceed the hardware capability.
    [Show full text]
  • A Parallel Sparse Direct Finite Element Solver for Desktop Computers
    CMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, Poland A parallel sparse direct finite element solver for desktop computers Sergiy Fialko 1 1Department of Physics, Mathematics and Computer Science of Tadeusz Kosciuszko Cracow University of Technology Warszawska 24 St., 31-155 Kraków, Poland e-mail: [email protected] Abstract The parallel finite element solver, which is based on block L·S·LT factoring, where S is a sign diagonal, is presented. The sparse symmetric positive definite and indefinite matrices are considered. Unlike the methods presented in the libraries of high-performance procedures, including PARDISO solver, the proposed approach uses the disk storage, if the dimension of the problem exceeds the capacity of random access memory RAM. Two out of core modes OOC and OOC1 are developed. The OOC mode produces the minimal I/O operations, ensures the stable speed-up when the number of processors increases, but requires respectively larger amount of RAM. The OOC1 mode demands the smaller amount of RAM, but the number of I/O operations is significantly greater than in OOC mode and speed up is much poorer. The presented method is essentially faster than multi-frontal solver, especially on multiple processors, because it is not using redundant data transfer from one memory buffer to another. Keywords: sparse direct solver, finite element method, parallel computing, multithreading, high performance, desktop computers OOC1 mode, which allows one to solve large problems on 1. Introduction computers with small memory, such as laptops. Currently, large engineering problems are increasingly 2. Analysis phase being solved on the desktop, rather than on workstations, clusters, and computer networks.
    [Show full text]